In this work, different from the resonance operator basis used in Ref. [1]}, we employ the ones proposed in Ref. [2]} to include the relevant odd-intrinsic parity operators beyond the minimal ones in the R\(\chi \) T framework. The merit is that the resonance Lagrangian in the latter basis has been widely exploited to investigate various physical processes and the relevant resonance couplings are also well determined [3]}, [4]}, [5]}, which can provide us valuable inputs for the study of the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  decay. To further take the on-shell approximations of the \(VJP\)  operators and use the \(\omega \rightarrow \pi ^0\pi ^0\gamma \)  decay width as an additional input, we are able to make parameter-free predictions for the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  decay. Therefore our study offers complementary results to the phenomenological aspects of the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  process, compared to the ones given in Refs. [6]}, [7]}, [1]}. Furthermore, another important novelty of our work includes the exploratory discussions of the T-odd asymmetry distributions in the \(\tau \rightarrow \pi \pi \gamma \nu _\tau \)  process, which can supply an important guide for future experiment measurements.
